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Contents 1. Algebra of Matrices ....................................................................................................... 1-76 2. Finite Difference and Difference Equation and Numerical Method .................... 77-121 3. Differential Equation ................................................................................................ 122-175 4. Function of Complex Variable ................................................................................. 176-235 1 ALGEBRA OF MATRICES 1.1 INTRODUCTION The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrix notation and operations are used in electronic spread sheet program for personal computer, which in turn is used in different areas of business and science like budgeting, cost estimation, analysing the results of an experiment etc. 1.2 DEFINITION If mn numbers real or complex or functions are arranged in the form of a rectangular array A having m rows horizontal lines and n columns vertical lines then A is called an m n matrix. Each of mn numbers is called an element of the matrix. There are different notations of enclosing the elements constituting a matrix viz , , but square bracket is generally used. An m n matrix is also called a matrix of order m n. For example, 2 1 5 3 0 4 is a 2 3 matrix or matrix of order 2 3. It has two rows and three columns. 1.3 DIFFERENCE BETWEEN MATRIX AND A DETERMINANT i A determinant has a numerical value but a matrix has no numerical value. ii The number of rows may or may not be equal to the number of columns in a matrix while in a determinant the number of rows is equal to the number of columns. iii Interchanging the rows and columns, a different matrix is formed while in a determinant, an interchange of rows and columns does not change the value of the determinant. 1.4 VARIOUS TYPES OF MATRICES a Row Matrix